On the Chandler wobble of Mars

Planet. Space Sci., Vol. 44, No. 11, pp. 1457-1462, 1996

Pergamon

Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0032-0633/96 $15.00+0.00 PII: SOO32-0633(96)00052-9

On the Chandler wobble of Mars V. N. Zharkov and S. M. Molodensky Institute of the Physics of the Earth, Russian Academy of Sciences, B. Gruzinskaja,

10, 123810 Moscow, Russia

Received 22 November 1995; revised 21 March 1996; accepted 22 March 1996

Introduction The Chandler wobble is the Eulerian free nutation of the planet. The wobble of the Earth was first observed by Chandler (1891), but was not recognized as the Eulerian free nutation because its observed period (434days) is much longer than the predicted period for the rigid Earth model (305days). The explanation of this disagreement was given by Newcomb (1892). The sources of the large discrepancy between the theoretical rigid body and measured periods of the Earth are : (1) the elasticity of the Earth’s mantle ; (2) the size of the liquid core and its flattening ; (3) the tidal motions in the ocean under the action of variable centrifugal force and (4) anelasticity of the mantle. First, elasticity results in a considerable lengthening of the rigid body period. In view of the elasticity of the mantle, the action of variable centrifugal forces on the Earth results in elastic deformations of its outer surface and in significant reduction of the Earth’s effective dynamical flattening entering Eulerian (Liouville’s) equaCorrespondence

to: V. N. Zharkov

tions. This effect causes the lengthening of the Eulerian period of the order of 143 days. Second, the liquid core of the planet causes a decrease in the period of the Chandler wobble. In view of weak coupling between the liquid core and the mantle, the vectors of angular momentum of the liquid core does not coincide with the vector of the angular momentum of the mantle, and consequently, the simplest Eulerian description is not complete. Effects of the liquid core for the real Earth model are of the order of - 50.5 days. Third, the Earth’s oceans cause an increase of 29.8 days in the Chandler wobble by reducing the effective difference between the principal moments of inertia of the Earth in the same manner as the elastic mantle. Fortunately, Mars has no oceans, so in the case of Martian wobble this effect may be ignored. Fourth, there is the loss of energy through anelastic dispersion in the mantle, that is, the change in the quality factor Q, with depth. This effect contributes a lengthening of the Earth’s Chandler wobble of 8-12 days. The problem of the Martian Chandler wobble is considered below, including the possibilities of determination of the aggregative state and the radius of the Martian core. Effects of the anelasticity of the mantle and the triaxiality of Mars’ figure are taken into account. To estimate the required accuracy of the measurements which is necessary for solving the problems under consideration, below we will estimate also the excitation of Martian Chandler wobble by atmosphere. This value may be considered as a lower limit of Chandlerian amplitude, because up to now the motions in the liquid core are not known.

The Eulerian period of weakly t&axial

Martian models

In the paper by Zharkov and Gudkova (1993a) the principal moments of inertia of Mars A
= (C-A)/B,y

= (B--)/C

(1)

1458

V. N. Zharkov and S. M. Molodensky: On the Chandler wobble of Mars

Table 1. The principal moments of inertia of Mars and their normalized differences (here the standard notations are used : M is the mass of the planet and R its mean radius)

B = co(C-

or 0 = o(C-2)/d.

Data Normalized moments and their differences

Balmino et al. (1982)

Smith et al. (1990)

C/MR= A/MR2 B/MR’ cr, 103

0.3678 0.3657 0.3660

0.3692 0.3671 0.3674

5.7148 5.0271 0.6877

5.7447 5.0553 0.6894

B, IO3 Y, IO3

are determined on the grounds of data of Balmino et al. (1982) and Smith et al. (1990) (see Table 1). In the case of the modelling of the planet by the ellipsoid of revolution the mean equatorial moment of inertia is determined by A = B(A+B)/2.

(2)

In these notations, the relation y < p ta holds, and the small parameter which defines the tri-axiality of the Martian figure is equal to 1.

As is shown below, the Eulerian period for the tri-axial Martian model is smaller than the same period for the two-axial model by 2 days. The free rotation of a rigid body is described by Euler’s equation of conservation of the angular momentum M : (4)

where D/Dt and the overdot are time derivatives in space and in the rotating reference frame, respectively, and cc)is the vector of the instantaneous angular velocity of the rotating reference frame. In the case of a rigid body it is convenient to match the axes of the mobile reference frame with the principal axes of the inertia tensor. In this case the relation between components of the vectors M and o is as follows : M,=Ao,;~~=Bw,;M,=Co,.

(5)

The substitution of (5) into equation following non-linear equations : dw,/dt + ao,o, do_Jdt-@,o,

(4) results

= 0

(7)

dw,ldt + yoYoY = 0.

(8)

A = B, y = 0, and equation

oz = 0 = const.

= -a~,,

dw,ldt

(8)

(9)

The substitution of equation (9) in equations results in the well-known Euler formula

where

(6)

= 0

In the zero-approximation is reduced to

dw,/dt

in the

= WJ,

(6) and (7)

A)(C-

B)/(AB))“2

= w(~$)“~.

Co,= c, cos(o,t) ; my = c, cos(o,t)

(13)

where c,/c,

= (a/P)“2.

(14)

For the Earth TE,E= 2nlo = 305 days.

For Mars, using equation T

(11) 187.6 days

E*M1= i 188.3 days

(Bahnino et al., 1982; Smith et al., 1990) and using equation (14) T

(Balmino

186.6 days E,M2= i 185.6 days

et al., 1982; Smith et al., 1990).

The Chandler period of weakly t&axial

Martian models

For the planet with an elastic mantle and a liquid core the Chandler frequency with a good accuracy can be written in the form (Molodensky, 1961) : -__ C-A

Bw-

A-J,

where A, is the moment Love number

(1-k1

of inertia

0

of the liquid core, k the

k. = 3G(C--A)/(w2R5)

which is the secular Love number, constant, and R the mean radius numerical values

1, we find

(16)

G the gravitational of Mars. Using the

T=2rc/~=8.8643x10~s,M=6.43x10~~g and data from Table

(15)

0

G = 6.672 x lo-* gg’ cm-’ SK’, R = 3.39 x lo8 cm, (10)

(12)

As distinct from the case of a two-axial rigid body, the trajectory of the pole with respect to the rotating tri-axial body is elliptical :

(3)

DM/Dt=i+I+oxM=O

(11)

Obviously, equation (9) is valid also in the case of an arbitrary tri-axial rigid body, if the amplitude of free nutation is small enough. In this case o,<< CL),,w,,<
y/cl = l/S<<

A)/A

V. N. Zharkov and S. M. Molodensky: On the Chandler wobble of Mars Table 2. The Love numbers k, moments of inertia of the liquid core A, (in 104’gcm2), periods of the Chandler wobble T, and T,, and corrections for the liquid core 6 T, and for tri-axiality of the figure 6T, Model

M512LL

M5121LS

0.130 1.07 203.9

0.083

P$ $;

202.8 202.7 199.7

194.9 192.0

;$: $ sd”

7.9 7.7 1.1 3

1.1 2.9

C-2

= 1.452 x 104’gcm2andk,

k

A,

C-A

= 1.465x

104’gcm2andk,

(18)

K(v,a) = K,(r,o)[ I+ iQx ‘@,a)].

(19)

Q;‘,QK1<
et al., 1982)

= 1.3040 (Smith et al., 1990).

The comparison of equations (11) and (14) shows, that the formula for the frequency of the Chandler wobble for a weakly tri-axial planet has the form 0 wl =

,4-d = ~~(r,d[l+ iQ; ‘@,o>l

The quality factor (or dissipative function) Q, (or QK) is related to the relaxation of the shear (the variation of the volume) processes. For rocks and the Earth’s interior

196.0

= 1.2921 (Balmino

1459

((C-A)(C--))“2 (&)1/2-A,

(20)

and therefore in planets without oceans the tidal delay and the damping of Chandler wobble are determined only by the distribution of QJr,o) ; &(Y,o) may be replaced by K,(r). The anelasticity of the mantle (and core, if it is solid) determines not only the value of damping of Chandler wobble, but gives also a significant contribution to the real parts of Love numbers and to the Chandler period (Zharkov and Molodensky, 1977, 1979; Smith and Dahlen, 1981; Molodensky and Zharkov, 1982). In the seismic range of periods z 1 s-l h, Q, is practically not dependent on the frequency. In the range of long periods (tides, Chandler wobble) there is the weak dependence of Q, from the frequency. The most direct way to obtain the weak dependence of Q, from the frequency is to use the power-law creep function 4(t) for the phenomenological description of anelastic processes in the high-temperature regions of Mars’ and Earth’s mantles

(17) A large number of trial models of Mars were constructed in Gudkova et al. (1993). These models are two(the core and the mantle) and five-layered ones (which include the crust of varying thickness, the upper olivine mantle, the transition zone from olivine to y-spinel, the lower mantle, and the core). For example, M512L denotes a five-layered model with a 150 km thick crust, a mean density of 7.0 gcmm3 for the core, and with a low temperature distribution. This model has been selected as the reference model in Gudkova et al. (1993), and we shall also use it here. The last letter for all the models is introduced to denote the aggregate state of the core : liquid (L) or solid (S). In Table 2 the second-order Love numbers k, the values T, and T,,, are the corrections for the tri-axiality of the figure (in days), 6T, = T,- T,,, and the corrections for the liquid core, 6T, = T,,(M512LL)T,,(M512LS) (in days) are collected. In Table 2 the values fld’ and T$f) correspond to the first and second columns of Table 1. We see from Table 2 that the effect of the liquid core is of the order of N 8 days, and the Chandler period of the tri-axial model is approximately l-3 days shorter than for the two-axial models.

The effect of anelastic behaviour of the Martian interior Phenomenologically, one may introduce damping into the theory of the anelastic isotropic medium by converting the real elastic moduli ,&r) and K,(r) to complex quanti ties :

where t is time, p the shear modulus, p a dimensionless constant of the material, E the energy of activation for micro-creep, k is Boltzmann’s constant, and T the absolute temperature. The power-law creep function describes the transient creep. The parameter M is bounded by the limits l/3 and l/2, but it may be somewhat smaller. For weak dissipation which is typical for planetary interiors the functions Q,(g) and ~~(a) have the form (Akopyan et al., 1977) : Q;’ = cnF(n)o-” &(O) = p( co)[ 1 - cnr@+-”

sin(nnj2)

cos(n71/2)]

= ,u( co)[ 1- Q,‘(a) where F(n) is the gamma frequency shear modulus. the basic formula

~ PO(@l)=

’- 2Q,(oJ

(22)

cot(nn/2)]

(23)

function, and ,U(CO)the highIt follows from equation (23)

[(o,/o*)n - l]cot(n71/2)

(24)

which can be used for the calculation of the Chandler period lengthening ST,. Because of non-elasticity, the Chandler frequency o, equations (I 5), (17) acquire a small negative variation Re(Go,) and small imaginary variation Im(Go,). It follows from equation (15) that these variations are connected with the corresponding variations of the Love numbers 6k by the formulae Re(Sa,)

c-d = - - A

2

0 -k Re(3 0

(25)

V. N. Zharkov and S. M. Molodensky: On the Chandler wobble of Mars

1460

Table 3. The quality factor Qw (equation (30)), and the relative lengthening of the Chandler period (6T,/T,,) x lo3 (equation (29)) for three values of the reference period T = 2n/a, T,, = 200 days

Table 4. The

values (6T,+,/T,,,,)x lo3 for the cases, when Q, = const. for 0 > dt, T,, = 200 days ?- @I

T (s) 200 1000 2 x 104

QW

0

0.05

0.1

0.75

0.2

0.25

0.3

6.9 5.9 4.1 550

7.3 6.5 4.8 390

8.0 7.3 5.8 280

8.9 8.4 6.9 200

10.1 9.7 8.4 140

11.7 11.4 10.2 100

13.7 13.5 12.3 70

Im(Go.J

= - A C-A 2

where Az is the mean equatorial Martian mantle for the models result, we get the formulae Re(Go,)

= - j&

kwIm(&) 0 moment of inertia of the with a liquid core. As a

K%/%>” -

II&

coto42) (27)

Im(6cW) = -

100 1000 2x lo4

0

0.05

0.1

0.b

0.2

0.25

0.3

6.9 5.9 4.1

9.2 6.5 4.8

12.7 9.9 5.8

17.8 13.1 6.9

25.5 17.7 8.4

37.1 24.0 10.2

54.8 33.1 12.3

At present, it is not known in what range of periods we must use equation (23) and in what range equation (31)? Undoubtedly, in the range of low frequencies cr< cr, equations (23) and (27) must be used. If at d >> ct the relation Q,(a > ct) = const. holds, then in the range of frequencies d 2 cr2 vt the values 6 T,,, are defined by equation (32). In Table 4 the corresponding corrections 6T, are larger than those presented in Table 3. It is known that for the Earth QWz 100. As it was shown in Zharkov and Gudkova (1993b), the mantle of Mars deviates more strongly from ideal elasticity than the mantle of the Earth. If we suppose, that for Mars QWz 100, then, in accordance with Table 3, the value 6T, is of the order of 2-5 days.

(28) The estimation of the amplitude of the Chandler wobble

= Qw=

$gKdQ ~

Re%

2Imo,

=

- 110

0

cot(nn/2)

55 1(ko - k)(o,o/oJ

(29)

(30)

which were used for the calculation of ST, and QW. In equations (27) and (28) ct is the reference frequency, and Tt2 = 21~10, = 2 x lo4 s the tidal period. The values 6T, as functions of the parameter y1are given in Table 3. If the reference frequency is in the seismic range of periods, for example, bzn x lop2 or 271x 10M3, then the corresponding corrections 6T, must be calculated by the formula 6T, = &

[(o/a,,>” - 113

cot(nn/2).

(29a)

The values 6T,,, and Q,,, (equation (30)) for several numbers y1are given in Table 3. As it was noted before, in the seismic range of the periods M 1 s-l h, Q, is not dependent practically on the frequency. In this case the logarithmic creep function must be used, and the expression for the dynamica shear modulus flO(d) is as follows :

The amplitudes of the Chandler motion of the Earth’s pole are determined mainly by the transfer of the angular momentum from the atmosphere to the solid Earth. These amplitudes can be found with the help of the equation of the total angular momentum conservation which includes the angular momentums of the mantle (M,), of the liquid core (M,), and of the atmosphere (iVIa). To get the simplest estimation, one can put M, z vpSR2

(33)

where u is the characteristic value of the averaged winds’ velocities, p the mean density of the atmosphere, S the area of the planetary surface, and R the mean radius. As it was shown in Zharkov and Molodensky (1994), the substitution of (33) into the angular momentum equation results in the following estimation : (34)

where A is the amplitude of Chandler wobble, C the moment of inertia of the planet, 6 the decrements of attenuation, and the subscripts E and M denote the Earth and Mars, respectively. Substituting in (34) the numerical values pM/pEz I x fOhZ; uM/uEz 1; R&/(R&)z2; ~M~~M/(&~~ = 1, and A Ez 100 mas (milliseconds of arc), we get A,/AEz2x 10-2, and A,z2mas. ~~(4= ~(~)[l+(2i(~Q,))(ln(~i~)+0.572)1. (31) The second possible source of Martian Chandler wobble excitation is connected with the irregular motions in For the rocks c1z lo3 Hz. The formula which is analogous its liquid core. In the case of the Earth, in view of the high to equation (24) has the form density of the atmosphere, atmospheric excitation exceeds effects of the liquid core significantly, and any attempts (32) to estimate the characteristic values of velocities in the

V. N. Zharkov and S. M. Molodensky:

On the Chandler wobble of Mars

liquid core using data about the variations of the Earth’s rotation are connected with significant difficulties. In the case of Mars the situation may be opposite. Then there is a good possibility to estimate not only the mechanical parameters of the Martian mantle and liquid core, but to estimate also the values of irregular velocities in its core. Obviously, such an estimation is of great importance for the theory of the magnetic field generation. The atmospheric effects estimation given above may be considered as a lower limit of the total Chandler amplitude of the real Mars which is excited both by the irregular motions in the atmosphere and in the liquid core. It is important to note that the precision of determination of the Earth’s Chandler period (z 1 day) is limited not by the precision of observations, but by the chaotic character of its atmospheric excitation. In view of this circumstance, even in the case of very small amplitudes of Martian free wobble, the accuracy of the Martian Chandler period determination may be comparable with the accuracy of measurements of the Chandlerian period of the Earth.

Discussion

and conclusion

The results given above show that the data on the Chandler wobble of Mars give essential information about the aggregative state of its interior and the radius of its liquid core. For the set of trial Martian models the corrections for the effects of the liquid core and anelasticity of the mantle to the period of Chandler wobble are of the order of 8 days ; the lengthening of the Chandler period due to the anelasticity of its interiors can reach ~2-5 days. In calculation of the principal inertial moments A, B, C (see Table 1) the well-known estimation of the mean moment of inertia I = 0.365MR2 was used, where M is the mass and R the mean Mars radius (Reasenberg, 1977 ; Kaula, 1979; Zharkov and Gudkova, 1993a). To determine the values A, B, C without any assumptions, it is necessary to add the value of the constant of precession H = C-V+@/2 c

(35)

which may be obtained also from the observations. After the works of Reasenberg and Kaula the value I = 0.365Mr2 was used practically in all publications devoted to the models of Mars, because it corresponds well to all data which are used usually in investigations about the aggregative state of Mars’ crust and mantle. In Bills (1989) some considerations are given, in accordance with those the mean moment of inertia may be significantly less (I- 0.345MR2). In the recent publication by Bills and Rubincam (1995) this problem is discussed in some detail. For this aim they draw the estimations of the speed of precession of Mars’ axis p =

- 3n2Hcos(Q) 2m(l_e2)332

(36)

where n is the orbital mean motion, Q the obliquity of the ecliptic, w the spin angular rate, and e the orbital eccentricity. All of the these parameters (except H, the

1461

precession constant) are known quite well. By using the data of Chandler and Reasenberg (1991) p = (-8.1*0.5)arcsecyr-’

and the data of Sinclair

and Morley

(1992)

p = (-7.85+1.62)arcsecyr-’

Bills and Rubincam obtained of inertia values as follows :

the ranges of the moments

0.3221CIM?
and Reasenberg’s

estimations)

and

0.293 I C/Mr2 IO.445 (for Sinclar and Morley’s estimations). It is well known that the value C/My2 must be smaller than 0.4, and it is clear that the accuracy of the determination of H is not enough to obtain accurate values of principal moments A, B, and C. As it was noted in Bills and Rubincam (1995), in the near future the precession rate will be known with the accuracy of the order of 1%, and after that the discussion about the value of I will be closed. If the principal inertia moments are known with a good enough accuracy, the problem about the aggregative state of the Martian core and the mantle’s anelasticity (in the range of periods of about 200days) may be solved by using the data about its forced nutation and Chandler wobble. The anelastic properties of the Martian mantle for the period 2 x lo4 s were obtained using the data about tidal delay of Phobos (Zharkov and Gudkova, 1993b). On the other hand, the amplitude of the semi-annual prograde nutational component is almost independent of anelasticity of the mantle, but it strongly depends on the aggregative state of the core (Groten et al., 1996). Thus, by joint analysis of the two types of data one can try to separate the effects of the aggregative state of the Martian core and the effects of the anelastic mantle at the period 200 days. The comparison of anelastic properties for the periods 2 x 104s and 200 days may be used to estimate the dependence of the dissipative factor in this range of periods. The amplitude of Mars’ Chandlerian wobble strongly depends on the values of irregular velocities in its liquid core. If Chandler wobble is excited by the Martian atmosphere, then this amplitude is of the order of a few milliarcseconds of arc, and registration of the Chandler period may be connected with rather strong technical difficulties. At the same time, if irregular (convective) motions in Mars’ liquid core are intensive enough, then the investigations of Chandler wobble can give important information not only about its internal structure, but also about dynamics of the liquid core. Acknowledgements. The research described in this publication was made possible by Grant Nos NFJ 300 (V.N.Z.) and NFE 300 (S.M.M.) of ISF and Russian Government and partly by Grant Nos 93-02-2856 (V.N.Z.) and 95-05-14231 (S.M.M.) of Russian Fund for Fundamental Research. We are also grateful to an anonymous referee for useful comments.

1462

V. N. Zharkov

References Akopjan, S. Ts., Zharkov, V. N. and Lubimov, V. M., Theory of attenuation of torsional oscillations of the Earth. Phys. Solid Ehrth No. 8, 3-l 1, 1977. Balmino, G., Moynot, B. and Volves, N., Gravity field of Mars in spherical harmonics up to degree and order eighteen. J. Geophys. Res. 87,9735-9756, 1992. Bills, B. G., The moments of inertia of Mars. Geophys. Res. Lett. 16,385-388, 1989. Bins, B. G. and Rubincam, D. G., Constraints on density models from radial moments: applications to Earth, Moon, and Mars. J. Geophys. Res. 100, 26,305-26,315, 1995. Chandler, J. F. and Reasenberg, R. D., Mars precession. Center for Astrophys. Smithson. Astrophys. Observ., Tech. Memo. 91-07, 11 pp. Cambridge, Massachusetts, 1991. Chandler, S., On the variation of latitude. Astron. J. 11, 83, 1891. Groten, E., Molodensky, S. M. and Zharkov, V. N., On the theory of Mars’ forced nutation. Astron. J. 1996 (in press). Gudkova, T. V., Zharkov, V. N. and Lebedev, S. A., A theoretical spectrum of the free oscillations of Mars. Astron. Vestn. 27(2), Y-75,1993. Molodensky, M. S., The theory of nutation and diurnal Earth tides. Comtnuns Ohs. R. Belg., 188 S. Geophys. 58, 25-56, 1961.

and S. M. Molodensky:

On the Chandler

wobble of Mars

Molodensky, S. M. and Zharkov, V. N., Chandler wobble and frequency dependence of Q, of the Earth’s mantle. Phys. Solid Earth 21, 167-175, 1982. Newcomb, S., Remarks on Mr. Chandler’s law of variation of terrestrial latitudes. Astron. J. 12,49-58, 1892. Sinclair, A. T. and Morley, T. A., The determination of the precession rate of Mars from Phobos and Deimos observations. Astron. Astrophys. 262,326-328, 1992. Smith, M. L. and Dahlen, F. A., The period and damping of the Chandler wobble. Geophys. J. R. Astron. Sot. 59, 223-284, 1981. Smith, D. E., Lerch, R. S., Nerem, G. B. et al., Developing an improved higher resolution gravity field for Mars. EOS Trans. AGU 71, 1427, 1990. Zharkov, V. N. and Gudkova, T. V., Parameters of the equilibrium figure of Mars. Solar System Res. 27, 103-l 10,1993a. Zharkov, V. N. and Gudkova, T. V., Dissipative factor of the interior of Mars. Solar System Res. 27, 299-309, 1993b. Zharkov, V. N. and Molodensky, S. M., Love numbers of anelastic Earth’s models. Phys. SoZid Earth 5, 17-20, 1977. Zharkov, V. N. and Molodensky, S. M., Corrections to Love numbers and Chandler period for anelastic Earth’s models. Phys. Solid Earth 6, 88-89, 1979. Zharkov, V. N. and Molodensky, S. M., On determination of physical parameters of the Martian core by data of its rotation. Solar System Res. 28, 86-97, 1994.